Antiferromagnetic (AFM) skyrmionium composed of two topological AFM skyrmions shares the merits of an AFM skyrmion, for example, high mobility and no skyrmion Hall effect. Here, we analytically and numerically study the dynamics of the AFM skyrmion and skyrmionium induced by spin currents. Our calculations demonstrate that the current-induced spin-transfer torques can drive AFM skyrmion and skyrmionium with the same speed, while their steady motion speeds induced by spin-orbit torques are different. Furthermore, it is found that due to the existence of the effective AFM texture mass, the AFM skyrmion and skyrmionium obey the momentum theorem, and the time evolution of the position induced by alternating currents presents a phase. Besides, a spin torque nano-oscillator based on the AFM skyrmionium can produce high frequencies, similar to that based on the AFM skyrmion. Numerical simulations are in good agreement with the analytical solutions. Our results demonstrate the inertial dynamics of the AFM skyrmion and skyrmionium and may provide guidelines for building skyrmion-based spintronic devices.